querying the universe: the dodd-jensen core model and ...nate/talks/berkeley/2009/querying...

Querying the Universe: The Dodd-Jensen CoreModel and Ordinal Turing Machines

Nathanael Leedom AckermanUniversity of Pennsylvania

Effective Mathematics of the UncountableCUNY

Aug. 20, 2009

Nathanael Leedom AckermanUniversity of PennsylvaniaEffective Mathematics of the UncountableCUNYQuerying the Universe: The Dodd-Jensen Core Model and Ordinal Turing Machines

Outline

(1) Review Of Ordinal Turing Machines

(2) Oracles And Absoluteness

(3) Query Machines

(4) Writing The Core Model

(5) Query Machines And Oracles (if time allows)


Ordinal Turing Machines

An Ordinal Turing Machine T consists of

Four ordinal length, 0 or 1 valued, tapes: Tape(Input),Tape(Output), Tape(Parameter) and Tape(Work).

State: A finite set of states including two distinguished states,Start and Halt.

Program: A function fromState × {0, 1}4 → State × {0, 1}4 × {left, right, neither}4.

Notice that an ordinal Turing machine is completely determined by〈State,Program〉 (which we call its code). We denote by OTMthe collection of all codes of Ordinal Turing Machines.


Run of an Ordinal Turing Machines

Given a code for an ordinal Turing machine T = 〈State,Program〉,a run of T consists of

For each tape Tape(i) a function Value(i) : Ord → 2Ord

A function Position : Ord → Ord4 such thatPosition(0) = 〈0, 0, 0, 0〉A function CurState : Ord → State such thatCurState(0) = Start

such that they are consistent with Program (in the standard way).

We want to think of these functions as taking the a time andreturning the condition of the machine at that time.


Run of an Ordinal Turing Machines on Input

A run of T on input I ∈ 2Ord with parameters P ∈ Ord<ω is a runsuch that

Value(Input)(0) = I

Value(Parameters)(0) = Characteristic Function of P

Value(Output)(0) = Value(Work)(0) = ∅

Notice that a run is completely determined by its starting valuesand the code for the ordinal Turing machine.


Writing Sets

Definition

A run writes a class S ⊆ Ord if it enters a halt state at a time tand Value(Output)(t) = S .

We say an ordinal Turing machine T writes a class S ⊆ Ord ifthere is a finite set of ordinals P such that the run of T with input∅ with parameters P writes S .

Our first observation is

Theorem

If T ∈ OTM writes S ⊆ Ord then S is a set.

Proof.

If a run halts at time t then the output tape is a subset of t.


Writing L

A natural question is

Question: What sets can be written?

Theorem

Every set that can be written by a T ∈ OTM is in L.

Proof.

Every code for an ordinal Turing machine is in L, every finitesubset of the ordinals is in L, and runs of an OTM on fixed inputare absolute between models of set theory.

Theorem (Koepke)

For every set S ∈ Powerset(Ord) ∩ L there is a T ∈ OTM suchthat T writes S


Writing Inner Models

This suggests the following definition:

Definition

Suppose M is an class inner model of set theory. Suppose GTuringis a collection of codes for “generalized ordinal Turing machines”(with ordinal tapes and time). We say that GTuring writes M ifthe collection of sets which are written by a T ∈ GTuring isPowerset(Ord) ∩M.

We now have the question:

Question: What class inner models can be written by more generalordinal Turing machines?


Oracles

Definition

Suppose A ⊆ Ord . A Ordinal Turing Machine with Oracle A is anordinal Turing machine with (another) distinguished tape,Tape(Oracle).

A run of such a machine is defined exactly as a run of an ordinaryordinal Turing machine except that Value(Oracle) is the constantfunction A. (i.e. the oracle tape always has A written on it).

A code for such a machine is a pair 〈T ,A〉 where T is a code foran ordinal Turing machine. We let OTM(A) be the collection ofcodes of ordinal Turing machines with oracle A

Notice that a run of an element of OTM(A) is completelydetermined by the input, the parameters, and the code.


Writing Sets From Oracles

Definition

We say that a set S is written by OTM(A) if there is an element〈T ,A〉 ∈ OTM(A), a finite set of ordinals P, and a run of 〈T ,A〉with input ∅ and parameters P such that the run halts with S onthe output tape.

Theorem

All sets which can be written by OTM(A) are in⋃α∈Ord L[A ∩ α] (= L[OTM(A)] = L[A])

Proof.

Suppose S is written by a run of 〈T ,A〉 ∈ OTM(A) withparameters P which halts at time t. Then S is also written by arun 〈T ,A ∩ t〉 ∈ OTM(A) with parameters P.


Writing Inner Models

Theorem (Koepke)

If A is a set then OTM(A) writes L[A].

Proof.

A slight modification of Koepke’s proof.

Corollary

For all class A ⊆ Ord, OTM(A) writes exactly those sets in⋃α∈Ord L[A ∩ α]

Corollary

For every class inner model M, there is a class AM ⊆ Ord suchthat OTM(AM) writes M


Absoluteness and Generalized Ordinal Turing Machines

While every model of set theory can be written by an ordinalTuring machine with some oracle, in general, the oracle needs tobe a class which equi-constructible with the entire universe.

This suggests the question

Question: Is there a more general notion of an ordinal Turingmachine (whose collection of “codes” forms a set C ) but whichwrites a class inner model larger than L[C ] (when one exists)?



Answer: No

...If we assume that runs of the machines are absolutebetween class inner models.

Proof: Every finite set of ordinals is in L[C ] and so every set whichcan be written in V by our general notion of Turing machines canbe written in L[C ] with the same run (as the runs are absolute).

In particular if we want to write a class inner model which is largerthan L with a generalized ordinal Turing which is contained in L,then our notion of computation must not be absolute.

Or put another way, our generalized notion of a Turing machinemust be able to “ask questions” about the ambient set theoreticuniverse.



Answer: No...If we assume that runs of the machines are absolutebetween class inner models.

Proof: Every finite set of ordinals is in L[C ] and so every set whichcan be written in V by our general notion of Turing machines canbe written in L[C ] with the same run (as the runs are absolute).

In particular if we want to write a class inner model which is largerthan L with a generalized ordinal Turing which is contained in L,then our notion of computation must not be absolute.

Or put another way, our generalized notion of a Turing machinemust be able to “ask questions” about the ambient set theoreticuniverse.


Set Theory With A Choice Function

In order to make precise the notion of a machine that can askquestions of the ambient universe we first must define which settheory we are using.

Definition

Let LST = {∈,V ,F} and let GBCF be the theory in LST

containing

The Axioms of Godel-Berney’s Set Theory.

{(x , y) : F (x) = y} is a class.

F : V → Ordinals is a bijection.

In particular our set theory satisfies the Axiom of Global Choicewith a preferred choice function.


Definition of a Query

Definition

We call a question that our generalized ordinal Turing machine canask of the universe a query.

Specifically a query is a formula (possibly with parameters) whichis the graph of a function from Powerset(Ord) to Powerset(Ord)in any class inner model of GBCF (containing the parameters).

Suppose ϕ(x , y) is a formula such that every class inner modelsatisfies (∀x)(∃y)ϕ(x , y). Then there is a query associated toϕ(x , y) given by:

ϕF (x , y)⇔ ϕ(x , y) ∧ (∀z)ϕ(x , z)→ F (z) ≤ F (y)


Pure/Non-Pure Queries

Definition

We say a query is pure, if the formula describing it is in thelanguage L∈ = {∈} (possibly with parameters)

Definition

We say a query is ∆Fn (A) if it is equivalent to a formula ϕF (x , y)

where ϕ(x , y) is a ∆n formula in {∈} with parameter A.


Query Machines

Definition

A Query Machine with query Q(x) = y is an ordinal TuringMachine with two extra distinguished tapes: Tape(QueryIn) andTape(QueryOut).

A run of such a machine is defined as a run of an ordinary ordinalTuring machine except that

(∀t ∈ Ord)Q(Value(QueryIn)(t)) = Value(QueryOut)(t)

A code for such a machine is a pair 〈T ,Q,A〉 where T is a codefor an ordinal Turing machine, Q is the query and A is theparameter used in Q. We let OTM(Q) be the collection of codesof query machines with query Q

A Query Machine with Query Q is such that Tape(QueryOut)always has the result of applying Q to Tape(QueryIn).


Runs of Query Machines

Definition

We say that a set S is written by OTM(Q) if there is an elementT ∈ OTM(Q), a finite set of ordinals P, and a run of T withinput ∅ and parameters P such that the run halts with S on theoutput tape.

Note a run is determined (in a fixed class inner model M) by itsstarting conditions as well as the code for the query machine.

Theorem

Suppose 〈Qi : i ∈ β〉 is a set size collection of queries. Then we cancreate a single query, Q∗, from which they all can be recovered.

Proof.

We can encode the sequence 〈Qi (x) : i ∈ β〉 as a single subset ofordinals in a uniform way. I.e. Q∗(x)(β · α + i) = Qi (x)(α).


Absolute Queries

Theorem

If Q is a ∆1(A) query then the collection of sets written byOTM(Q) in V is the same as the collection of sets written byOTM(Q) in L[A].

Proof.

If x ∈ L[A] then QL[A](x) ∈ L[A]. But because Q is a ∆1(A)function it is absolute. Hence QV (x) = QL[A](x). So any run of anelement of OTM(Q) in V is the same as that in L[A].

Hence if we only use ∆1(A) queries we can never written a modellarger than the one containing the collection of codes for the querymachines.


Writing the Universe

If we allow ∆F1 queries instead of just ∆1 queries, then we loose

absoluteness.

Theorem

There is a ∆F0 query QV such that OTM(QV ) writes V .

Proof.

Let ϕ(x , y) be the statement “If x is treated as a subset ofPowerset(Ord) then y 6∈ x” and let QV (x) = y ⇔ ϕF (x , y)

So QV (x) = y if y is the F minimal element of the universe not inx . With QV we can recover F−1(Ord) ∩ Powerset(Ord) byrepeated queries. Hence as F−1(Ord) = V every set of ordinals inV can be written by OTM(QV ).

Notice in particualar that OTM(QV ) is in L even thoughOTM(QV ) writes V .


Writing Intermediate Models

We have seen that query machines can write L as well as write theentire universe. This suggests the question:

Question: Is there a query Q (with OTM(Q) ∈ L) which can writea class inner model strictly between L and V (under someconditions)?

Answer: Yes. Further, we can find a pure query QDJ such that theOTM(QDJ) always writes the Dodd-Jensen core model.

But before we do this lets give a brief review of the core model.


Outline of the Core Model Below a Measurable

Definition

A Dodd-Jensen Mouse is a transitive model M = JUα such that

(i) U is a normal κ-complete iterable M-ultrafilter on some κ < α

(ii) All iterated ultrapowers of M by U are well-founded

(iii) M = HM1 (γ ∪ p) (the Σ1 Skolem hull) for some γ < κ and

some finite p ⊆ α

Definition

The Dodd-Jensen Core Model is KDJ = L[{M : M is a mouse}]


Important Property of Mice

For our purposes the most important properties of mice is

Theorem

For every mouse M there is a minimal pair (βM , λM) with βM anordinal and λM a cardinal such that

L[M] = L[JCλMβM

]

(where CλMis the closed unbounded filter on λM)

Corollary

KDJ = L[{JCλMβM

: M is a mouse}]


Writing the Core Model Below a Measurable

Theorem

There is a pure query QK such that OTM(QK ) writes KDJ in anymodel of set theory.

Proof.

Let QK (x) = y be the formula which says

Case 1: If x = 〈α, β, λ〉 then

y = Lα[{JCλMβM

: βM < β, λM < λ and M is a mouse}] ∩Powerset(Ord) ordered by the canonical ordering of L

Case 2: Otherwise

y =⋃

x


Writing the Core Model Below a Measurable

Proof.

It is then clear that in any class inner model QK (x) = y is thegraph of a function from sets of ordinals to sets of ordinals. HenceQK is a pure query.

It is also clear that that every set in Powerset(Ord) ∩ KDJ can bewritten by a query machine in OTM(QK ).

However, if S can be written by a machine OTM(QK ) which halts

at time t then S must be in L|t|++ [{JCλMβM

: |βM |, |λM | < |t|+ and

M is a mouse}]. Hence S must be in KDJ and OTM(QK ) writesKDJ .


Query Machines and Oracles

Theorem

For every A ⊆ Ord there is a ∆0(A) query machine QA such thatOTM(QA) writes the same sets as OTM(A)

Proof.

Let Q(x) = A ∩⋃

x

Theorem

For every Σn(A) query Q there is a Σn(A)V oracle OQ such thatOTM(Q) writes the same sets as OTM(OQ)

Proof.

Let OQ encode all output of all the query machines run with Q


Query Machines and Oracles

Theorem

There is a ∆F0 query Q such that for every model M |= GBC and

every A ∈ M there is an expansion of M to a model MA |= GBCFwhere OTM(Q) (in MA) writes the same sets as OTM(A).

Proof.

Let ϕ(x , y) = (⋃

x) · 3 ≤ y ≤ (⋃

x) · 3 + 1 and letQ(x) = y ⇔ ϕF (x , y).

Next let be any bijection such that if x ∈ Ord

F (x) = γ · 3 when x ∈ A

F (x) = γ · 3 + 1 when x 6∈ A.

Given a machine in OTM(A) we can find a machine in OTM(Q)which mimics it (when run in MF ). Likewise, given a machine inOTM(Q) (run in MF ) we can find a machine in OTM(A) whichmimics it.

Hence the sets which can be written by OTM(A) are exactly thosesets which can be written by OTM(Q) in MA.


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